$\mathsf {dL}_{\iota }$: Definite Descriptions in Differential Dynamic Logic

Abstract

We introduce , which extends differential dynamic logic () for hybrid systems with definite descriptions and tuples, thus enabling its theoretical foundations to catch up with its implementation in the theorem prover . Definite descriptions enable partial, nondifferentiable, and discontinuous terms, which have many examples in applications, such as divisions, nth roots, and absolute values. Tuples enable systems of multiple differential equations, arising in almost every application. Together, definite description and tuples combine to support long-desired features such as vector arithmetic.

We overcome the unique challenges posed by extending with these features. Unlike in , definite descriptions enable non-locally-Lipschitz terms, so our differential equation (ODE) axioms now make their continuity requirements explicit. Tuples are simple when considered in isolation, but in the context of hybrid systems they demand that differentials are treated in full generality. The addition of definite descriptions also makes a free logic; we investigate the interaction of free logic and the ODEs of , showing that this combination is sound, and characterize its expressivity. We give an example system that can be defined and verified using these extensions.

This research was sponsored by NDSEG, the AFOSR under grant number FA9550-16-1-0288, and the Alexander von Humboldt Foundation.